Maybe I finally figured out how to input a grid, but there must be an easier way (I hope). I've looked at this thing til my eyes hurt. I then erased everything, hoping a fresh start might be better, but it wasn't. If someone could just give me a nudge, perhaps I can take it from there. I have another puzzle that's similar, so maybe the same nudge will help on both.

By the way, I have been able to solve only four cells: r2c7, r5c5, r6c6 and r8c7.

Notice that there are only two ways to place a "5" in row 1. There are also only two ways to put a "5" in columns 4 & 6, and in the top center 3x3 box. These possible "5"s form a binary chain ... I've marked them with "+" and "-" signs to accentuate this relationship. Evidently all the "5+" cells are 5, or else all the "5-" cells are 5 in the final solution.

Now look at row 5. There are also only two ways to place a "5" in this row, at r5c2 or at r5c9. I've marked these "5~" and "5=" to distinguish them from the "+/-" marks we made earlier.

Now we can draw some inferences.

If r1c9 = 5 then r5c9 <> 5, and r5c2 = 5
If r1c9 <> 5 then r8c4 = 5 (r1c9 is a "-", and r8c4 is a "+")

So either r5c2 = 5 or else r8c4 = 5, and there cannot possibly be a "5" at r8c2. dcb

To type a grid in by hand, use NotePad (on Windows). It uses a fixed xx font, so columns will line up properly.

Then, in the forums: Type your message. When you are ready to put in the grid, click the "code" button, then cut and paste from notepad. Click the "code" button again.

Otherwise, get some software (like Sudoku Susser) that allows you to easily type a puzzle onto a grid, or it reads many formats. Then, you can output the puzzle as a text file at any stage through the solution.

I'm not sure what it was either, except I was having trouble figuring out how to input a grid. I finally clicked "Code", then typed my numbers, with the appropriate number of spaces so the columns would line up, then clicked "Code" again. I'll try the Notepad and see how that goes.

Quote:

These possible "5"s form a binary chain

Is a "binary chain" the same as a "forcing chain"? I just learned of the latter a couple of days ago, but haven't had an opportunity to use it yet. I figured that I could use it when a bunch of cells had just two candidates.

How did you discover that chain? What were you looking for?

I have two other unfinished puzzles similar to the one I posted here. Both have a fair number of cells with four or five candidates and my current repertoire of techniques doesn't cut the mustard. Chances are, they will also require the use of chains similar to the above.

It can be but need not necessarily be so.
Binary chains can be either uni-directional or (less
commonly) bi-directional in terms of "forcing".

The purer form of the binary chain has the property
of being a single sequence of cells but there is also
the alternative kind known as "double implication"
chains - where the path of the chain depends upon
the values in the cells "visited" on the path.

A "binary chain" is any which consists at each link
with a choice of exactly TWO routes or values to be
considered in order to determine the next link. If
the "incoming" route to a cell determines the route
of the "outgoing" link then the chain is a "forcing" one
at that point and if ALL the links are of that nature then
the whole chain is a 'forcing chain'.

Sometimes a chain is forcing in one direction or with
one set of values but not with the alternative.

eg a line with 67 68 79 where the route arrives at 67.

If the cell has value 6 then the next cell is 79 (not 6)
and if it has value 7 the next cell is 68 (not 7).
(the chain is forcing irrespective of the 67 value).

However if the route arrives at 79 the situation is not
so clear. Arriving with 7 forces a link to 68 (not 7) but
arriving with 9 leaves the next link as either 67 or 68
and so the chain is not forcing - but it is still a binary!

> How did you discover that chain? What were you looking for?

This is a complex subject. There have been a number of
conjectures as to how to narrow the search for chains but
there has been no definitive method agreed yet. Each case
is open to inspection but those using DIC's over recent months
have learned some intuitive tricks as to where the chains are
more likely to be. They then need testing. Occasionally, a chain
exists and is valid but does not contain any useful information!

The "gold standard" for a chain is where it can be demonstrated
that cell A can have two (occasionally more!) values and that
whichever value is held in that cell the value in cell B is BOUND
to be a unique value.

For example A=67, B=89 and chains can be found such that A=6
forces B to be 9, and also that taking A=7 forces B to be 9. This
is not the norm. In most puzzles the chains from one binary pair
(eg A=67) will lead to different values in cell B.

Some human solvers will find first a "reductio ad absurdum" or
'logical contradiction' so that assuming one of the two values for
a particular square will lead to a breach of the Sudoku prime rule
that each row/column/region must contain all digits 1-9. This is
taken as a sign that the OTHER value must be correct (although
usually tested to be so!). It is good form to recast this into the
preferred logic of having disparate values in one cell leading to a
unique value in another.

Having said all this, the puzzles in the Daily Sudoku have rarely (if
ever!) required the use of these chains. However the "other puzzles"
section of the forum contains numerous examples where chains have
been needed to reach a solution. My suspicion (!) is that sometimes
the human solver has used trial and error to determine the solution
and then worked backwards to find the chain - but I may be wrong!

The use of chains is in the "ultra-advanced" group of techniques and
only recently have the computer solvers been enhanced to find them
(sometimes with large time-resource overheads). My belief is that a
human being needs a particular penchant for searching for chains as
the mental discipline is beyond that of simple logic - or even of the
search for patterns in the candidate profiles. One needs to keep track
of implications from cell to cell across many cells- rather than just
spotting relationships in a 'static' situation. However that does not
deny that some puzzles, proven to have a unique solution, can be
solved only with chaining techniques.

Many human solvers will be attracted by the difficulties of being on
the "leading edge" of solution methods. However, no-one NEEDS to
aspire to that role. The Daily Sudoku provides each day a puzzle that
can challenge the human without recourse to such "ultra-advanced"
methods. For example, it is usually possible to solve a "medium"
puzzle without making ANY pencil marks (except the solution of
course!) and the Hard/V.Hard can often be solved using marks only
for the Mandatory Pairs rather than using Candidate Profiles.